We have the following property
\(t(YY^{\prime })=\mathrm {max}\lbrace t(Y),t(Y^{\prime })\rbrace ,\) 
which is a consequence of the isomorphism with the bigraded Tor
\(\widehat{\mathit {HM}}_{*}(YY^{\prime })=\mathrm {Tor}^{\mathbb {F}[[U]]}_{*,*}(\widehat{\mathit {HM}}_{*}(Y),\widehat{\mathit {HM}}_{*}(Y^{\prime })).\) 
and the fact that, forgetting about the gradings,
\(\mathrm {Tor}^{\mathbb {F}[U]}_{*,*}(\mathbb {F}[U]/U^{n},\mathbb {F}[U]/U^{m})&=\mathbb {F}[U]/U^{n}\oplus \mathbb {F}[U]/U^{n}\text{ for }n\le m\\\mathrm {Tor}^{\mathbb {F}[U]}_{*,*}(\mathbb {F}[U],\mathbb {F}[U]/U^{n})&=\mathbb {F}[U]/U^{n}.\\\mathrm {Tor}^{\mathbb {F}[U]}_{*,*}(\mathbb {F}[U],\mathbb {F}[U])&=\mathbb {F}[U].\) 

This follows (with a little homological algebra over the principal ideal domain \(\mathbb {F}[U]\) ) from the description as mapping cone of Bloom, Mrowka and Ozsváth (see [1]}) or alternatively from the formula in Heegaard Floer homology [2]} via the isomorphism between the theories (see [3]}, [4]} and subsequent papers). Because also \(t(\bar{Y})=t(Y)\) , we have that \(t(X)\le N\)  for any \(X\)  obtained by connected sums of elements of our collection (with multiplicity and either orientation). The result then follows from Proposition REF  and the computation (REF ).
